Re: When the universe was smaller was the geometry of space closed?

Excellent question! First, let me note that the density of the universe determines two fundamental properties: the geometry and whether it will expand forever. In the absence of dark energy (see below), there’s a “critical” density at which the universe will (just barely) expand forever and is spatially flat. A density smaller than this value means that the universe is open and will expand forever; a density larger than this value means that the universe is closed and will eventually contract. The geometry and expansion are equivalent in the simple case of a universe with just normal matter. It’s more intuitive to think about the expansion, so let’s begin with that.

Now to the analogy. Suppose there’s a wind coming from something very massive, like the sun. The wind will expand into the space around it. As it expands, it will also slow down because of the gravitational attraction between the wind particles and the sun. If the wind began fast enough, it would be able to escape completely. If it began slow, the wind would eventually stop and get trapped. So there’s some “critical” initial velocity at which the wind particles will just barely be able to escape the sun.

Let’s suppose we observe the wind at some distance from the sun’s surface (like near the earth), and we want to figure out if it will escape to infinity. We can’t compare the wind’s speed at this point to the critical initial velocity mentioned above. This is because, by the time the wind has reached the earth, it’s been feeling the sun’s gravity for quite a while and will have slowed down from its initial speed. But of course it is also pretty far from the sun’s surface, so at the earth’s orbit, it doesn’t need to be traveling quite as fast to escape eventually. The point is that the numerical value of the critical velocity depends on where we evaluate it.

How does this apply to the universe? The wind is just an analogy for the expansion of space, as you probably noticed. The critical velocity for the wind is similar to the critical density. The difference is just that, in this case, the matter providing the gravitational attraction is everywhere, and it expands with the “wind.” We want to ask whether the gravitational attraction from each part of space is enough to stop the expansion. As the universe grows, the expansion decelerates because of the gravitational attraction between each particle and those around it. But the density also decreases, so the gravitational attraction becomes less significant. It turns out that the two effects exactly cancel. The density of matter decreases with time, but the critical density decreases at the same rate. So the ratio of the two remains constant with time: if the universe has the critical density at any one time, it will have the critical density at all other times. In terms of the wind analogy, if a particle has precisely the velocity needed to escape the sun at some point, it will have exactly the right velocity at every other point too. Because the critical density also determines flatness, that means that if the universe is flat now, it always has been (and always will be).

There is one significant complication: the real universe has dark energy, which behaves differently from normal matter (in some ways it behaves like antigravity). This means that the flatness gets decoupled from the eventual fate of the expansion: geometrically, dark energy acts just like normal matter, but it also helps the universe expand. So even though we now believe the universe to be very nearly flat, the dark energy means that it will continue to expand (at an accelerating rate!) forever.

You might find the following website useful: it has a much more extensive (but still reasonably non-technical) discussion of these issues and more:
http://www.astro.ucla.edu/~wright/cosmolog.htm